Gamblers like bets with high potential gains. A bookmaker could accommodate this preference by simply offering better odds. Alternatively, bookmakers could more cost-effectively offer high potential gains by nudging customers to bet on more complex and specific events. (In gambling I define “complexity” as the number of relevant events that could possibly happen, which need to be considered in order to make a rational judgment.) My first research question was, do they?

Yes. My research investigates the advertisements posted in TV and betting-shop windows by British bookmakers for soccer. I find that bookmakers seldom advertise bets on broad events such as “Manchester United to win.” Instead, bookmakers tend to advertise bets on more specific sub-events, such as “Manchester United to win and both teams to score” or “Manchester United to win 2-1.”

Why are complex gambles priced so unfairly and advertised so frequently?

Why? The more specific an event, the less likely it is to occur. This statistical fact means that a bookmaker can offer better odds. In principle, this holds the bookmaker’s profit margin constant; in reality, bookmakers price bets so that complex gambles produce the highest profit margins. (This article explains bookmaker profit margin calculations.) Bets on specific score lines—for instance, “Manchester United to win 2-1”—produce average margins of 23.2 percent. Bets on the identity of the first player to score average 34.6 percent. My second research finding is that the simplest gambles, such as “Manchester United to win,” produce the lowest profit margins, averaging 5.7 percent.

So why don’t consumers notice the premium price of complex gambles and shift their purchases to simpler events offered at fairer prices? A naive preference argument would say that perhaps consumers do notice but are blinded by the high potential wins offered by complex gambles. Or perhaps consumers pick complex gambles because they misperceive the odds of winning. I conducted a series of experiments to find out.

In the experiments, soccer fans were asked to forecast the probability of each possible event, which represented varying degrees of complexity. Soccer matches are one-off events; one interpretation of statistics claims that probabilistic claims cannot be made for one-off events. Therefore, I looked only at how much a participant’s sum of probability judgments exceeded 100 percent. For example, if Manchester United is playing against Chelsea, then the probabilities of the three possible outcomes (Manchester United wins, Chelsea wins, or the teams draw) must logically sum to 100 percent. This also holds when breaking the events of a soccer match into more-complex sub-events. A bet on “Manchester United to win and both teams to score” doubles the number of potential key events to six (each of the three events can happen one of two ways). The sum of these six events must again logically equal 100 percent. A sum of probability judgments exceeding 100 percent is called “incoherent;” incoherent bets means that a gambler can be persuaded to accept a string of bets that are guaranteed to make her poorer.

With the British public’s annual gambling losses exceeding $16 billion, there is increasing public awareness that policy has allowed the bookmakers too much freedom.

According to this standard, soccer fans are very bad at estimating complex event probabilities. For the simplest gambles, like “Manchester United to win,” the average participants did OK, giving judgments that summed to just over 100 percent. On more complex gambles, like “Manchester United to win and both teams to score,” participants gave mean judgments sums of up to 166.7 percent. For first-goal-scorer events, participants were even worse, giving mean judgment sums of up to 248.8 percent. And worst of all, for score lines, participants gave mean judgment sums of up to 306.5 percent for the 16 most likely score lines.

This research doesn’t conclusively rule out alternative explanations for why British bookmakers advertise complex events to consumers. But it does show that the average soccer fan lacks the skills to accurately judge complex (but not simple) gambles. This in turn can help explain why complex gambles are priced so unfairly and advertised so frequently. With the British public’s annual gambling losses exceeding $16 billion, there is increasing public awareness that policy has allowed the bookmakers too much freedom. However, the public are largely unaware of gambling advertising complexity.

For benevolent choice architects, the best way to improve consumer welfare might be to remove complexity-increasing dark nudges.

Philip Newall is a postdoctoral researcher in engineering psychology at WMG, University of Warwick and a former professional poker player. He is interested in how principles of behavioral science affect the marketing of gambling and financial products.